Luận văn Wavelet và ứng dụng trong nén tín hiệu video

z)=Go(z)H 0(z) = H 0(z)H1(-z) detHm(z) (dodet(H",(z))=-det(H",(-z))) 52 ( X(z) J]Ho(-z)] X(-z) (3.2.9) II II (3.2.11a) (3.2.12) " II 3.Khaitriln tinhi~udungb(j l(Jc -2 GJz)H] (z)= Ho(-z)H] (z)=P(-z) detHm(z) (3.2.10)cothec1uqcvie'tl~i: P(z) +P(-z) =2 (3.2.13) V~y,ne'uPia mQt?athuctheozthlP coH(tcacach<%sf)chanb~ng0,trup[O]=1. dodo,P(z)cod~ng: P(z) = 1+L:p[2k +1]z-(2k+]) kel MQtc1athucho?c mQthamhuuti thoa(3.2.13)c1uQcgQi la hqp 1<%(valid).Ham P(z) dongmQtvai trc)quail trQngtrongvi<%cphantich va thie'tke'cac bQ lQc.Cong thuc (3.2.10)co thec1uqcvie'tl~ila : Go(z)Ho(Z)+Go(-z)Ho(-z) =2 vie'tdudid~ngcuahamc1apling xung,hayd khiac~nhthaigianla : L:go[k]ho[n- k] +(-I)nL:go[k]ho[n-k] =2J[n] kel kel 2L: go[k]ho[2n - k] =2J[n] kel hay (go[k],ho[2n- k) =J[n] vdiqJk,qJkc1uqcc1inhnghlanhu(3.2.4)va(3.2.6),c1~ngthuclIentrdthanh: (qJO,qJ2n)= J[n] tu'clngtV SU'd~ng(3.2.10 - 3.2.11)taseco : (qJj,f{J2n+]) = J[n] (ifto,rP2n+])= 0 (ift] ,f{J2n+])= 0 VI({>lk,C{Jzk+1l~nluqt la phienb~mtinhtie'ncua f{Jo,f{J1nencach<%thucco thec1uqcvie't l~i: (iftk'f{Jz) =J[ k -I] V~ydieuki<%nkhoi ph~cloanhaDkeDtheoc1ieuki<%nsongtnfc giao.Dao l~i,ne'uC?P cdsa {qJk,iftk}suydien tu ho,hI, go,gIla songtnfc giao thl bQ lQc la khoi phvcloan haG. Nhu'v~yb~ngcachxemxet lIen khia c~nhc1ieuche'tinhi<%uta co clingke'tqua nhu khixemxet lIen khia c~nhthai gian : ndu ki~nkh6iph{lcloan hao tu:angdu:angwii dduki~nsongtri!Cgiao . 1.3Phantichtheokhfa c~nhdapha Phiintichtheokhia c~nhc1ieuche'cho mQtbieu dien tV nhien nhungco mQtnhuqc di€mladuthua.Trangmatr~nH1ll(z),cach<%sf)xua'thi<%n2 l~nVI ca lQcHi(z)va phienbanc1ieuche'Hi(-z) cuano clinghi<%ndi<%n.MQtcachbieudiengQnhdnla phan tichtheokhiac~nhc1apha.Ta se phanfa ca tin hi<%uva lQcthanhcac thanhph~nc1a pha.Dungcongthuc (2.4.11)vdi N =2 c1ebieu dien tin hi<%uke'tqua sail khi lQCva la'ym~uxu6ng. 53 3.Khai triln tin hi~udung bQ 19C ( Yo(z) ] = ( Hoo(Z) HOl(Z) ] ( Xo(Z) ]~(z) HIO(Z) Hl1(Z) X1(z) ' v ' ' , ' v ' Y(z) H v(z) X (z) p p TrangdoHi}la thanhphc1ndaphathll j cuaIQcn, vabieudi€n cuaHi rheacacthanh phc1ndapha,rhea(2.4.10- 2.4.11)la : Hi(z)=HiO(l) +ZHil(l) Y(z)la tinhi<$ucida~lllgiuacuah<$th6ngtranghlnh(3.4a), Hiz) lamatr?nvoicaethanhphc1ndaphacuacacIQcthanhphc1n,congQilamatr(in dapha. . Xp(z)g6mcaethanhphc1ndaphacuatin hi<$unh?pXova Xl, tinhi<$unh?pdu'<;1cbieu di€n rheacacthanhphc1ndaphabCii(2.4.8) X(z)=XoCl) +Z-lxl(l) Hlnh(3.4b)chi ra so d6 kh6i cua (3.2.14).Tin hi<$uX du'<;1cta hthanhhai thanhphc1n daphaXovaXl dungbiend6i dapha.Ma tr?n daphaHpbiend6i2 tinhi<$unh?PXO,Xl thanh2tinhi<$uxua'tYO'yI. (3.2.14) ~ ,.~ Yo ~t + R2t Z.IJ\ (a) Yo x Hp YI (b) Gp £ Yo YI (c) ffinh 3-4 Phiin tich tIeDkhia Cl;lnhdapha (a) Bie'nd6i dapha di toi va di lui. (b)Phftnphiin tich.(c)phftnt6nghcJp. Phfint6ngh<;1ptranghlnh(3.4a)clingdu'<;1cphanrichrheacachtu'ongtV.No co the du'QCthvchi<$nnhophepbiend6idaphangu'<;1c,matr?ndaphat6ngh<;1pdu'<;1cdinh nghial : 54 3.Khai triin tinhi~udungb(J lr;c ( Goo(z) GlO(Z) ] Gp(z)= GOI (z) Gll (z) Trongdo Gi(z)=Gw(l) + Z-IGil(Z2) (3.2.16) Trangcacthanhph~ndaphacuaIQct8ngh<;5pdu'<;5cdinhnghlavoi phangu'<;5cvoi cac IQcphantich.Tin hit$usankhi t8ngh<;5pX(z) du'<;5cviSt la : ( Goo(Z2) GlO(Z2) J ( Yo(Z2) JX(z)=(1 Z-l) GOl(Z2) Gll(Z2) ~(Z2) (3.2.15) (3.2.17) I '--v ' G (Z2) Y (Z2) P p Tin hit$utu mlii kenhdu'<;5cIffym~uleu b6i 2, ta du'<;5cYi(Z2),sando du'<;5cIQ b6i Gi(z)=Gw(l) +Z-IGil(l) voihaithanhph~ndaphaGwva Gil.PhepIffytichmatri;ln Gpva thanhph~netaphay la theoZ2,dodoco th6du'<;5cthl/chit$ntru'ockhi Iffym~u tenb6i2,nhu'6h'inhve(3.4c) BQIQcphantichvabQIQct8ngh<;5plad6ing~ucuanhau.BQIQcphantichdungphep biSnd8i daphadi Wi, pha 1du'<;5clamnhanhmQtnhip,ky hit$ub~ngchITztrongsod6 kh6ih'inh(3.4b).BQIQct8ngh<;5pdungphepbiSnd8i daphadi lui, pha1du'<;5clam tr~ (delay)mQtnhip,ky hit$ub6iZ-I trangvangtrail 6 nhanhdu'oih'inh(3.4c). HQp2 phepbiSnd8i daphaphantichva t8ngh<;5p,tadu'<;5cbQIQcphantich,t8ngh<;5p comatr(mchuyinddi dapha (transferpolyphasematrix)la : Tp(z)=Gp(z)Hp (z) Tinhit$usankhi quabQIQcphanticht8ngh<;5pla : A ( l\r. 2 2X(z) =1 z- pp(z )y(z ) =(1 Z-1 )Gp(z2)Hp(Z2)Xp(Z2) =(1 Z-1 )rp(Z2)Xp(Z2) DiSukit$nkh6iphvcloanhaocuaht$th6ngla : X(z) =X(z) khiTp(z)=I, taco : X(z) =(1 Z-l) ( Xo(z:) ) =XO(Z2) +Z-lXj (Z2) =X(z) X1(z) V~yTp(z)=I ladiSukit$ndudeht$th6nglakh6iphvcloanhao. 11.1.4Quanh~giii'acaebiiu di~ntheoth«igian,di~uche'biende)vadapha. Bi€udi~ntheothaigian,diSuchSbiendQva daphala 3 cachnh'inkhacnhaucua clingillQtht$th6ng,v'ivi;lychungco lienht$qual~il~nnhau. VlX(z)=~(Xo(Z2)+Z-IX1(z)) (tu(2.4.8))tacoth6viSt:2 ( XO(Z2)j 1 ( 1 J( 1 1 )( X(z) JXl(Z2))=2 z 1 -1 X(-z) (3.2.18) 55 3.Khai triflntinhi?udungb(3irc Nhu'v~y,tathie'tl~pdu'Qcm6i lien hl$giuabi€u di€n dapha X p(Z2) =(xo(Z2)XI(Z2)r vabi€u di€ndi€u che'biende) X"'(z)=(X(z)X(-z)Y hayI I cuatinhil$u. Xet lQcphanrich,taco m6i lienhl$giuaHp(z)vaH",(z)nhu'sau: ( Hoo(Z:) Hol(Z:) ] =! ( Ho(Z) HI(Z) ]( 1 1 )( 1 -I JHlO(Z) H1\(z) 2 Ho(-z) H](-z) 1 -1 Z 2 1 ( 1 1 J( 1 J Hp(z ) =-H m(z) -I2 1 -1 Z tu'ongW',taco : ( Goo(Z2) GOI(Z2) J =~ ( 1 J( 1 GlO(Z2) G1\(Z2) 2 Z 1 Gp(Z2) =~( 1 J( 1 1 J Gm(Z) 2 Z 1 -1 Caecongthlic (3.2:19),(3.2.20)la chuy€n vi cuanhau. (3.2.19) 1 J( Go(z) G](Z) )-1 Go(-z) G](-z) hay (3.2.20) Dethie'tl~pm6iquailhl$giuabi€u di€n rheathaigianvadapha,taxetcaclQct6ng hqpgi.Trenkhiaqnh thaigian,matr~nt6nghQpTscod~ngnhu'(3.2.5).D?tTsCz)la matr~ntu'onglingvoimatr~nTsdinhnghlanhu'sau: k'-] T,(z) =LSiZ-i ;=0 trongdo,Sila matr~nca'p2x 2 cod~ng(k' la chi€u dailQc) ( g [2i] g [2i] J - S = 0 I i =0 k'-1 1 go[2i+l] g][2i+l] , V~y: -r,(z)= k'-l Lgo[2i]z-; ;=0 k'-I Lgo[2i +l]z-; ;=0 k'-] L gl [2i]z-i i=O k'-] LgI[2i +l]z-i ;=0 k'-] L goo[i]Z-i =I i=D k'-] Lgo] [i]Z-i ;=0 k'-] LglO[i]Z-i i=O k'-] L g1\[i]z-; i=O = ( Goo(z) GlO (Z) JGOl (z) G1\ (z) TsCz)= Gp(z) V~yTlz) clingchinhla matr~ndaphaGp(z).D~ngthlictrenchatam6ilienhl$giua 56 3.Khaitriln tinhi~udungb(j Irc bieudi~ntheothai gianvdi matr~nTsva bieudi~ndaphavdi ma tr~nGp(z).T11'ong tv,matr~nphantichtheothaigianTaco lien ht$vdi matr~nphanitchdaphaHp(z) K Ta(z)=L:Aiz-i i=O KIa ehieudailQephantich,A lamatr~nea'p2 x 2 A = ( ho[2(K- i) -1] ho[2(K - i) - 2] JI hi[2(K -i)-I] hi[2(K -i)-2] K-I L:ho[2(K - i) -1]z-i T (z) =I i=1a K-I L:~[2(K -i)-I]z-i i=1 K-I L:hol [K - i]Z-i =1 i=1K-I L:~I[K -i]Z-i i=1 = K-] z-KL:hOl[K -i]ZK-i i=1 K-I Z-K L:hll [K - i]ZK-i ;=1 K-I L:ho[2(K -i)-2]z-i i=1 K-I L:~[2(K -i)-2]z-i i=1 K-I L:hoo[K -1- i]Z-i ;=1 K-I L:hlO[K -1- i]z-; ;=1 K-I Z-K+IL:hoo[K -1-i]ZK-I-i ;=1 K-I Z-K+IL:hlO[K -I-i]zK+i ;=1 =Z-K+I ( Z-IHol (Z-I) Hoo(Z-I) ]z-IHll(Z-I) HIO(Z-I) =Z-K+I ( Hoo (Z-I) HOl(Z-I) )( 0 HIO(Z-I) Hll(Z-I) Z-I To(z)=Z"'H,Vto, ~) d~ngthuetIeDthietl~pm6ilienht$gifi'amatr~nphanitchtheothaigianTavdi ma tr~nphantichdaphaHp(z). 11.2Caetinhcha'tcuabe)IQc: Trangphgnnay,tatoml11'9tcaeketqualienquaildencaebQlQesad\lngcaee6nge\l vuathietl~p,caeketquahoikhaehonsovdieaeke'tqualienquaildenslft6nt~ieua cdsatrveehu§'nhoi;iesongtnfegiao.d day ta quail tam den slf khoi phl:lcgdn dung, vanha'nm<;lnhvaonhfi'ngva'ndelienquaildenthietkelQe. Trangcaephgntrude,dieukit$nkh6iph\leloanhaoconghlala xua'tlit$ubiingdung nMplit$u.d daytat6ngquathoavadjnhnghlakh6iph\leloanhaonghlalaxua'tlit$u coth€ lamQtphienbanlamtn~,vacothed11'9CnhantYlt$sovdinh~plit$u: X(z)=cz-kX(z) Voicaeketquad ph§ntrudc,tatha'ydieutIeDt11'dngdu'dngvdi, quamQtpheptinh ~) 57 3.Khaitriln tinhifit dungbQlQc tie'nvanhanty l~,thlcachamdaplingKungcuacaclQCphanrichvalQctanghcjpt~o thanhmQtcdsasongtnfcgiao. Trongs6 cach~th6ngthoakhoiphlJ.cxa'pxi, taquailtamde'ncach~th6ngtranh du'cjcalias.TrangmQtbQlQcphantich/ tanghcjpbie'ndaitheothaigian,xua'tli~ula hamCllacii x[n]vaphienbandieuche'(-ltx[n] (hayX[z]va X[-z] trongmienz). Nhu'v~y,lo~iboaliasla mQtrangnhii'ngva'ndequailtrQng. Djnhnghia3.2 Matr~nca'pn x n Fij(z)du'cjcgQila matr~ntVayang(pseudocicularmatrix)ne'uco d~ng: { F. .(z) J .?:.i F (z)= O,J-I IJ Z.Fo,N+f-;(z) j <i Nhu'v~y,matr~nt(tayangtu'dngtvnhu'matr~nYang,nhu'ngami'atamgiacdu'oi,cac s6h~ngdu'cjcnhanthemz. M~nhd~3.4 TrangmQth~th6nglQcphantich/ tanghcjphaibangchotinhi~umQtchien,aliasse du'cjclo~ibone'uvachine'umatr~ndaphaTplamatr~ntvaYang. Changminh.. Giii si'tTiz) Ia matr~nh,tavong,Tp(Z)co d1;lng: ( Fo(Z) ~(Z» ) T (z)= p z~(z) Fo(z thayvao(3.2.18)taco : X(z) =(1 Z-I)rp(Z2)Xp(Z2) =(1 Z-l (FO(Z2)Z2~(Z2) ~ (Z2))[Xo (Z2»)FO(Z2) Xl(Z2) =(Fo(Z2) +zF;.(Z2) F;.(Z2) +Z-IFo(Z2){X 0(Z2» )\Xl (Z2) =(F(z) Z-lF(Z)(XO(Z: »)X1(z) =F(z)(Xo(Z2) Z-IXl(Z2») =F(z)X(z) V~yM th6nglabfftbie'ntheothaigiandodolO1;liboalias. H" ?~qua: Ne'uh~th6ngla lo~ibo alias,thldieuki~ndn vadud€ khoiphlJ.cloanhaGla ma tr~nchuy€ndaidaphala ffiQtmatr~nyanglamtIe,ho~ctremQts6channhip2k. k ( l O JTp(z)=cz- o 1 ho~cmQts6l6nhip: T (z)=CZ-k-l ( O 1 Jp z o . 58 3.Khai triin tin hi~udung bQ [flC Changmink: BS kh6iph1,1cto~mhilo,taco : X(z) =cz-k-IX(z) 2 2 ~ hay: F(z) =Fo(z ) +zF;(z ) =cz- v~y: FO(Z2)=CZ-k', Fl(Z2)=O hoi[Lc:F; (Z2)=CZ-k'-l, Fo(Z2)=0 ne'u: FO(Z2)=CZ-k', k'chan,tacok'=2kva Fo(z)=cz-k ( l'o(Z) 0 ) ( I 0 )khido: Tp(z)= =CZ-ko l'o(z) o 1 vaM th6ngla lamtrc~2knhjp: X(z) =CZ-2kX(z) 17 ( 2 ) -k'-l F] Z =cz ne'u: k'-I chanhayk' =2k+l, F; (z) =cz-k-] khido,matr~ndaphala : , ( OF, (Z) ) ( 0 Tp(z) = zF1(z) 0 =C Z-k vah~th6ngla lamtr~2k+1nhjp: X(z) =CZ-2k-1X(Z) Z;-l) =CZ-k-{~~) . M~nhd~3.5 rhomQtbQ19C2 kenhH(ymiluxu6ngbdi2 voi matr~ndaphaHp(z),khi do,kh6i ph~clakh6ngcoaliasneuvachineuHp(z)codinhthUGkhac0,noicachkhac,Hp(z) cohc;mgdli la 2. Changmink: Ch<;m atr~nt6ngh<,1pla matr~nph1,1h<,1pcuaHp(z) Gp(z) =cofactor (H(z» taco: Tp(z)=Gp(z)Hp(z)=del (Hp(z».I hiSnnhien Tp(z)la matr~nt1!aYang,dodo, co aliasdu'<,1clo;:liboo daDl;:li,ne'uM th6ngkhOngalias,khidoTp(z)lagiayangnencoh;:lngdula2. SuyraHp(z)coIwngdula2. M~nhd~3.6 rho mQtbQ19Cphantlch FIR, di~uki~nkh6i pht).cloan hao voi 19CFIR du'Qcthoa ne'uvachineudel (Hp(z» la mQtpheplamtr6thu§nlily. Changmink: GiasU'Hp(z)lami)tphept011nlamtr~thuiintuy,chQnbi)IQct6ngh<,1pco Gp(z) =cofactor(H(z» tmhiSnnhientacokh6iph1,1cloanhilovoi IQcFIR (theom~nhde3.5).Bao l;:li,ne'u tacodi~uki~nkh6iph1,1cloanhilovoiIQcFIR tmTp(z)lagiayangtinhtie'n,nghiala ( I 0 ) k ( O I )Tp(z)=Z-kO 1 hOi[LcTp(z)=Z--IZo trongca 2 tru'ongh<,1p,taco : . 59 3.Khaitriin tinhifjudungb(j lClc det(Tp(z»)=det(Gp(z))det(Hp(z»)=z-I vllQc t6ngh<,Jpcling Iii IQcFIR, det(Gp(Z»chi co th~co nghi~mho~cco Qic t(;li0, vii det(Hp(Z))clingv~y. Suy fa det(Hp(Z))la mQtpheplamtn~thu1lntuy. Giua(Hp(Z»va matri;lndieuch€ Hm(z) co m6i lien ht%bdi (3.2.19) 2 1 ( 1 1 J( 1 J Hp('z ) =-Hm(z) -]2 1 -1 z suyra : del (Hm(Z»=-zdet(Hp(l» vi;ly :del (Hm(Z»co d;;mg: del (Hm(Z»= a Z-2k+l Vi;lyn€u det(Hp(Z»khongconghit%mngoai0,del (Hm(Z»clingvi;ly. N6uht%th6ngla khoi ph\lc loan haGkhong co lam tr~,khi do, Hi,Gj =0,1lien ht% rheaGongthuG(3.2.11a): ( Go(Z) J 2 ( H] (-Z) JG](z) - detHm(z) -Ho(z) Go(z) =~Z2k+]H] (z)a G](z)=_~Z2k+]Ho(-z)a . (3.2.21) Nhu'vi;ly,tube)h/cphantichFIR, taKaydvngdu'9cbe)h/ct5ngh9Pdeht%th6ngphan rich/ tangh9Pla khoiph\lcloanhaGkhongbi lamtr~.Tuy nhien,loi giai la khong nhanqua.Decoloigiainhanqua,tanhanGo,G1voithuaso'Z-2k-l,nhu'nght%th6ngla kh6iph\lcloanhaGlamtr~2k+1m~u. Tu'dngttfnhu'trongbiendi~ndapha,k€t qualIenveIO(;libi)aliasclingco du'9ctrong lanhvtfcdieuch€. Tinhit%uxua'tquaht%th6ngsankhiphantich/ t5ngh9Pxettrong lanhvtfcdieuch€ la : 2(z) =~(Go(z) G](z){Ho(z) Ho(-Z» )( X(z) )2 \H](z) H](-z) X(-z) I ( X(z) ) =-(Go (z) G](z»)Hm(z) 2 X( -z) V~yc1ieukit%nde IO(;libi)alias(nghlala thanhph§n aliasX(-z) khongt6nt(;litrongk€t quaxua't)lavectordong(Go(z) G] (z»)Hm(z)cothanhph§nthuhaib~ngO.Hay: { Go(z)Ho (z) +G](z)H] (z) =F(z) (3.2.22) Go (z )H 0(- z) + G] (z )H 1(- z) =0 Nhu'vi;ly,matri;lnTm(z)=Gm(z)Hm(z)co d(;lng , ( F(Z) J Tm(Z)= G,nCz)Hm(z)= F(-z) MQtrongnhungWigiaicho(3.2.22)du'9CdenghibdiCroirier,Estenban,Galandla lQcQMF(quadraturemirrorfilter),trongdo,aliasdu'9clo(;libi): 60 3.Khaitriln tinhi~udungbQ19C [ HJZ)=HO(-Z) Go(z)=Ho(z) Gj(z)=-Hj (z)=-Ho(-z) Nghic$mtrenthoadi~ukic$nGo(z)Ho(-z)+GI(z)HI(-z)=0,dodo,aliasdu'Qclo(;liboo D~cokhoiphvctO~lllhaodi~ukic$nc§nvadula X(z) lake'tquatn~cuaX(z). ~ lr ]x I X(z)=-LGo(z)Ho(z)+Gj (z)Hj(z) (z)=z- X(z)2 hay Go(z)H0(z)+Gj(z)Hj(z)=2z-1 vdilQcQMF,di~ukic$ntrentrathanh H;Cz)- H;(-z) =2z-1 (3.2.24) VdilQcFIR, d€ tha'yr~nglQcHaarthoadi~ukic$n(3.2.24).Ngu'oitachlingminhdu'Qc r~nglQcHaarla lQcFIR duynha'thoa(3.2.24) Tatomt~tcacke'tquav~bQlQcsongtnfcgiaob~ngdinhly sail: DinhIy 3.1 TrangmQtbQlQchaikenhsongtnfcgiao,cacphatbi€u saildayla tu'dngdu'dng: a) hi[-n], g}n-2m])=bli-Jl bln-m] b) Go(z)Ho(z)+Gj (z)Hj(z) =2 va Go(z)Ho(-z)+Gj(z)H](-z)=O c) T.f.Ta=Ta.Ts=I d) GI1I(z)HI1I(z)=HIIl(z)G,nCz)=21 e) Gp(z)Hp(z) =Hp(z)Gp(z) =I J CaebQIQctqic chu§n Trangph~n aytaxetcacbQlQcthoacacrangbuQcsail: a) T(;lo<facd satnfcchuftncho12(Z),nghiala cactinhic$utrong12(Z)co th~khaitri€n trl;l'chuftntheocachamcdsasurdi€n tii'bQlQc. b) BQlQclaFIR. X6tbQlQcphantfch/ tanghQphaikenhvoicachamdaplingxungho,hI,go,gIthoa gi[n] =hi[-n] i =0,1, taxaydl;l'nghamcdsa {(jJk}nhu'sau: (fJ2k[n]=go[n-2k] (3.2.25) (fJ2k+I[n]=gI[n-2k] (3.2.26) Nhu'v~yhamdaplingxungcualQctanghQpla phienbanduonguqcthdigiancua hamdaplingxungcualQcphantich,vacachamcdsala phienbantinhtie'nmQts6 channhipcuahaihamdaplingxungcualQctanghQpgo,gl. d~t qJk=(jJk khid6di~ukic$nd~{(jJk}lamQttnfcchuftntu'dngdu'dngvoidi~ukic$n{qJk,(jJk}la mQt cosatrt1cgiao. , Ne'uM {(jJdt(;lOthanhmQtcdsatnfcchuftn,tanoibQ19Cla trlfcchudn. .1Trt1ehuintronglinhvl,l'cthO'igian VdibQlQcphantfchtanghQptangquatthoadi~ukic$n(3.2.25)va cachamcd sa (3.2.23) 61 III""""" 3.Khaitrain tin hi?u dung bi?19C dliqcdinhnghIabdi (3.2.26),dieDkit%n{lpdt?o thanhme)tcdsd tntcchuffntrdthanh: < gi[n-2k],gAn-21])=bIz-)] bIk-l] vacdsdd6ing~u{ifJk}trungvdi{lpk}nen< hi[n-2k],hAn-21])=bIi--:j]bIk-l] A ~ h 1,matr?n tong <,5pa: --" =TTa TaTs =TaT;=I V~ydieDkit%nlIen tu'dngdu'dngvdi Ta la ma tr?n Unita. N6i cach khac, dieDkit%n thaiphvcloanhaocuame)tbe)IQctntcchuffntu'dngdu'dngvdi matr?nphantichTa Ia matr~nUnita. b~ngcachd~tHi Ia thanhphgntu'dngling vdi kenh i cua be)IQctrongmatr?n phan tichTava Gi Ia thanhphgntu'dngling vdi h~nhi cuabe)IQctrongmatr?nt6ngh<,5pT,s, taco: G- =HT1 1 SHYra : HjH; =5[i - j]I iJ =0,1 V~ytinhit%usaDkhi du'<,5clQcphantichHi'ym~uxu6ng,Iffym~uIen r6i IQct6ngh<,5pd 62 go[O] gIrO] 0 0 go[1] gl[I] 0 0 go[2] gl[2] go[0] gIrO] T =I'" go [3] gl[3] go[1] gl[I]s I'" go[L-I] gl[L-I] go[L-3] gl[L-3] ... 0 0 go[L-2] gl[L-2] ... 0 0 go[L-I] gl[L-I] ... ho[O] hI[0] 0 0 ho[-1] hI[-1] 0 0 ho[-2] hI[-2] ho[O] hI[0] T =I'" ho[-3] hI[-3] ho[-1] hI[-1]s I'" ... ho[1- L] hI[1- L] ho[3- L] hI[3- L] ... 0 0 ho[2-L] hI[2-L] ... 0 0 ho[1-L] hI[1-L] ... 3.Khaitriln tin hi~udung bl}[flC kenhi se du'Qcbi~udiSnb~ngloantd'Mi Y. =Mx=HTH.x1 I I 1 MT =Mva M2 =M1 1 1 1 dodoMi la loanHi'chie'utntcgiao,nhu'v~y,bQlQchai kenhtu'ongli'ngvdi phepchie'u tnfcgiaoxuo'nghaikhonggiancontu'ongli'ngsinhbdicaehamcosd{lpzk}va{lpzk+l} tuongli'ngvdikenh0vakenh1, Dieukit$nkhoiphl,lcloanhllOla : H~Ho+H;H1 =1 BieudiSnTll,Tsdu'did~lllgcacmatr~nkho'ivdiAla matr~nca'p2x 2,dieukit$ntnfc chu5ntrdthanh: K-l" ATA =IL... 1 1 taco 1=0 K-I" AT A =0,L... 1+] 1 ;=0 j=I,...,k-l ,3.2Tqfc chufi'ntrong Ianh v1;icdi~uche' 'firdieukit$ntnjc chu5n,taco : <go[n],go[n+2m])=b'[m] (3.2.26) vip[l]=<go[n],go[n+l])la day tu'ongquailcuaday {go[n]}tC;licacchi so'chanl =2m, d~tp'[m]=p[2m]lala'ym~uxuo'ngbdi2cuap,taco: P'(z)=~[P(ZI/2)+P( -ZI/2)]2 P'(Z2)=~[P(z)+P(-z)]2 vifez)=Go(z)Go(Z-l)nen(3.2.26)trdthanh ~[Go(z)Go(Z-I) +Go(-z)Go (_Z-I)] =12 hay: Go(z)Go(Z-l) +Go(-z)Go (_Z-l) =2 tudngtl1vdi: <go[n],gl[n+2m]) =0 <gl[n],gl[n+2m])=b'[m] G1(z)G](Z-I) +G](-z)G1(_Z-I) =2 GO(z)G1(Z-I) +GO(-z)G1(-Z-I) =o Teenvongtroll donvi, (3.2.27a,b)trdthanh: IG; (elaJr + IG; (el(aJ+1r) r =2 V~ybQlQctnjc chu5nthoadieukit$nt6ngbinhphu'ongbien dQb~ngh~ngso',congQi ladieukit$nSmith-Barnwell.Dieu kit$nlIen duQcdungd~thie'tke'bQlQctnjc chu5n. Viet(3.2.27- 3.2.28)du'didC;lngmatr~n,taco : ( Go(z-]) Go(-z-] ) J( Go(z) G](z) ) = ( 2 0 )G1(Z-1) GJ_Z-l) Go(-z) G](-z) o 2 (3.2.27a) taco: (3.2.27b) (3.2.28) (3.2.29) 63 3.Khaitriin tinhi?u dungb(J19c hay G~(Z-I)Gm(Z)=21 VIhilaphienb~mdaongu9cthaigiancuagilienht$thuclIencoth~dU9Cvie't Hm(Z-I)H~(z)=21 Matr?nthoadieukit$n(3.2.29)dU9CgQila matr?nParaUnita.Ne'ucacthanhph~n cuamatr?nla6ndinh,matr?nlIendU9CgQilabaaloanthongtin(lostless). Matr?nchuy~nd6ila baaloanthongtinthltu'angduangvoibQlQct<:t°ramQtphep bie'nd6itnfcgiao.Vi lQctadangxetla lQcFIR lienmatr?nlabaaloanthongtin. Tir(3.2.27)taco : (G](Z-I )G1(- Z-I)Ytn1cgiaovoi(Go(z)Go(- Z-I)Y. C6 th~chungminhdU9C: G1(z)=_Z-2k+lGO(_Z-I) hay: gI[n]=(-IYgo[2k-l-n] V?y trongmQtbQlQc2 kenhtnfcgiao,m9i19Cdtu du(/csuydiln titmQt19Cduynh{lt. Ngoaifa, doFez)+P(-z) =2 lien lQcphaico chiendai la mQts6ch~n(b6de 3.1). (3.2.30) ,,3Tn;icchu!introng lInh v1;icdapha Matr?ndaphalien ht$voi matr?ndieuche'bdiht$thuc(3.2.20) Gp (z2)=~ ( 1 J( 1 1 ) Gm(z)2 z 1 -1 Dodo, G;(Z')Gp(Z')~~G:(Z-l{: ~r =~G~(Z-l)Gm(Z)2 V?y,ne'ubQlQcla tn1cgiaothl: GJ (Z-l)Gp(Z)=1 tacfingco: Gp(z)GJ(z-1)=1 VdibQlQcphantichcomatr?ndapha: Hp(z)=GJ(Z-I) thl: Gp(z)Hp(z) =Hp (z)Gp(z) =I V?ybQlQctn1cgiaothoadieukit$nkhoiphvcloanhaGkhongtr6nhip. 1.4TomHitcacke'tquavi:bQlQctr1;ichu!in: Dinhly 3.2: TrongmQtbQlQcFIR tnfcchuffnht$s6thlfc,tacocacke'tquasan: a) <gi[-n], gAn+2m])=b[i-j] o[m] ij =0,1 b) Go(z)Go(Z-I)+G1(-Z)GO(-Z-l)=2 va G1(Z)=-Z2k+IGo(-Z-1) c) T,TT,=1'.,1'.,T=I, Ta =T,T d) G~(Z-l)Gm(z)=Gm(z)G~(Z-I) =21,H m(z)=G~(Z-I) z-T J: ~l)Gm(Z) 64 3.Khai trifi'ntin hi~udung b(J lr;c e) G;(Z-I)Gp(Z) =Gp(z)G;(Z-I) =I, Hp(z)=G;(Z-I) LQcFIRtnjcgiaocochiendai1amt)tsO'chillinhub6desan: B6d~3.1 BQ1Qchaikenhtnjc chuffn,FIR, ht%s6th1fCfuoacacHnhcha'tsan: a) Chien dai L cuabt)1Qc1achilli , hayL =2K. b) Bt) 1Qcthoa dieu kit%nSmith-Barnwell ve t6ng blnh phuongbien dt) trenvangtrail donvi. IGo (eim r +IGo (ei(m+JT)r =2 IG1(eimr+IG1 (ei(m+;r)r =2 Ne'u1Qcbangtha'pco nghit%mti;lin nghla1aGo(-1) =0thl Go(1)=J2 LQckenhcaoco th€ duQcSHYdi€n tu 1Qckenhtha'pb~ngcachdao nguQcthaigianvalamtr€ mt)ts616nhip. G1(z) =_Z-2k+1GO(_Z-I) trongdo,2K=L 1achiendai1Qc. Chungmink: a) c) d) Khongmit tinht6ngquat,co th~gilt su cac M s6 cua 19CgoIii go[O],.., go[L-1] di;Lt: p[l]=(go[n],go[n+l])ladaytu'dngquailcuaday{go[n]} thl: p[L-l]= Lgo[n]go[n+L-l]=go[O]go[L-l]:;i:O nEZ b) Mi;Ltkhac,biSnd6izcuap laP(z)=Go(z)GO([I)thoa P(z) +P(-z) =2 do do, cacM s6 b~cchilli cuaP(z) pMi bhg 0, (tn)'p[O]=1). SHYrap[L-1] la hi!?s6b~cIe, hayL chilli Ta cotli (3.2.27a)vdiz=ejm, VIcacM s6cuagoIii thvclien Go(-elm)=Go(eim) taco' Go(elm)Go(ejm)+Go(-elm)Go(-e jm)=2 IGo(ei"r +IGo(ei(mT=2hay tu'dngtv,dung: Co(z)Co(-z) + C1(z)C1(-z) =2 vdi z=eim taco: IGo(ei"f+IG\(ei"f =2 Tli(b),vdiw=0,tacoIGo(lf+IG\(-lf=2,dodo Go(l)=.J2 Xemtrong[1]. c) d) M~nhd~3.7 TrangmQtbQ 1Qctnjc giao, Hang1uQngduQcbaa roan: Ilx112=IIYoI12+IIYll12 Chungminh: NangItiQngcuatinhii!?uxuitlIencackenhla : 65 3.Khaitriln tinhi{!udungbr?19C IIYol12+llyJ2 =~ nYo(ej&r+I~(ej&r~w27r0 (Yo(Z»)Y(z) = =H (z)X (z),, Y,(z) p pvoi taco: ~ f~Yo(ei& ~2 +IY,(ei& ~2~w=~ f[Y(ei&)]"Yo(ei&flw 27r 0 27r 0 =_ 21 f[xp (ei&)]*[Hp (ei&)]"Hp (ei&)xp (ei&flw7r 0 1 2. . . =- f[x (eJ&)jx (eJ&}iw 27r 0 p p = IIxo112+IIxl1l2= Ilxlf (VIHp(z)lEimatr~nParaUnita) . III CAC Be>LOC CO cAU TRUC CAY: BQlQcnhiSukenhco thedu'QCthie'tl~ptli'bQlQc2 kenhb~ngcachl?p l?i vit%ctach tinhit%u(j cackenhcan.MQttru'onghQpthongdvngla l?p l?i vit%ctachtinhit%u(j kt~nht<1nsatha'p.TagQibQlQcnaylabQlQCbatdQ(octavebasefilterbank). x giai dO1)1l2 "~ ~ (a) giai dOI)Il.T ; giai do,!1l1 .. giaidol)ll2 giai dol)l1.T (b) Hlnh 3-5 BQ h.>cbat dQquaJ giai do~n,voi phan fa thanhcac kh6nggian con tntc giao. ~=~+1EB~+1'Ne'uHi[n] la illQt lQc tntc giao vdi gi[n] =hi[-n]ca'utrucbatdQtfent~ofa khaitri6n tntcchuifn chu6iWaveletfCiif~C.(a)Phiin phantich.(b) Phiin t6nghqp. III.! B(ilQCbatdOvachu&iWaveletrO'ir~c: X6tbQlQcnhu'(j hlnh 3.5ta tha'ytin hit%udu'Qcphantachthanh2 nhanhqua IDQtbQ 66 3.Khaitriln tinhi~udunghi?l(Jc IQc2 kenh,saud6ph~ntinhit$ud kenhthffpIf;liduQctachlamhaivdi cungmQtbQ IQC,va cli tie'pt~c.Gia sabQIQcthoaDietlkit$nkhaiphl,lCtoanbaa,tasethffycffu trucnaycaid?tchu6iWaveletroi rf;lcsongtntcgiao.Ne'ubQIQcla tntcchu§'n,tase duQchu6iWaveletroi rf;lctntcchu§'n. Trangca2 tru'onghQpsongtntcgiaova tntcchu§'n,ht$cosd duQcsuydi€n titcac hamGaplingxungcuabQIQct6nghQp.VI V?ytase t?PtrungvaobQIQct6nghQp. Vi dl:l: Xet IQc Haar vdi cac ham Gapling xungdinhnghlad (3.1.9)cac bie'nd6i z tttongungla:Go(z)=~(l+Z-)~ G)(z)=~(l-Z-)) I Taxetquatrlnht6nghQpnhudhlnhve (3.5b),quaJ =3giaidof;ln,nhuV?y,tadung3 bQIQchaikenh.Vi IQcbdiG(z)saud6Iffymfiulenbdi2thltu'ongduongvdi Iffymfiu lenbdi2 nSiIQcbdi G(l) lienquatrmhtrentuongduongvdi bQIQc4 kenhvdi cac IQc: G(1)(z)=G(z)=~ (l-z-) ) ) ) 12 G?)(z) =Go(z)G)(z2)=~(I+Z-)_Z-2_Z-3)2 G?)(z) =Go (z)Go (Z2)G) (Z4)= If,. (1+z-1+Z-2+Z-3 _Z-4 _Z-5 _Z-6 _Z-7)2...;2 G~3)(z)=Go(z)Go(Z2)Go(Z4)= If,. (1+z-) +Z-2+Z-3+Z-4+Z-5+Z-6+Z-7) " 2...;2 duQcthltchit$nsailkhi tinhit$udil duQcIffymfiul~nluQtbdi2,4,8,8. CaehamGaplingxungduQcrutrab~ngcachIffybie'nd6iznguQc. Takyhit$ug6k)[n]la hamGaplingxungtuonglingvdi quatrlnhIQCk giaidof;lnd bangthffp,m6igiai dof;lnduQcthltchit$nsaukhi tinhit$uduQcIffymfiu!enbdi 2. g;k)[n]lahamGaplingxunglingvdiIQcbangcaotheosaubdik-I giaidof;lnIQc bangthffp,dm6igiaidof;lndeliduQcthltchit$ntru'dcb~ngphepIffymfiuIenbdi2. ) 2./2(1,1,1,1,1,1,1,1) 1 2 (I, I, -I, -1) . 1 2./2(1,1,1,1,-1,-1,-1,-1) Hlnh 3-6 BQh?cbatdQtangh(jpvdilQcHaarvaqua3giaidoliln. Trangvi d~tn3n,tac6 : 67 3.Khaitriln tinhi~itdungbQlQc 2 2 k G63)(Z)=Go(z2)G62)(Z)=flGo(22 ) k=O 2 2) k G?)(z)=G) (Z2 )G~2)(Z)=G1(Z2 )flGO(z2 ) k=O hayt6ngquat: J-) k G6J)(Z) =flGo(Z2 ) k=O (3.3.1) J-2 (J) 2J-) fl 2kGj (z)=G) (z ) Go(z ) k=O vih~th6ng1atnfcchu§'n,lienmatr?nphantichTavamatr?nt6nghQpT.51achuyen vi clla nhau. Ma tr?n phan tich bien di6n bdi cac ham dap U'ngxung h;1)[n],h;2)[n],h;3)[n],h63)[n],g8mcacdongchU'acach~so'cuacachamtren,Ta co d~ng: (3.3.2) AO Ta= AO Matr?nnay rho tha'y1Qcg?)[n] du'Qcthtfchi~nsankhi 1a'ym~u1enbdi 2 (dongsan du'Qcdichph,h2 cQtsovdidongtru'dcdo), g)2[n]du'Qcthtfchi~nsankhi1a'ym~u!en bdi4, gj(3),g63)du'Qcthtfchi~nsankhi 1a'ym~u1enbdi 8. Xettheokhia qmh thaigian,xua'tli~ucuah~th6ngd hinh (3.5a)trentungkenhdu'QC viet1a: HpH)L-)X j =1,2,...,J-1 vaakenhcu6irung 13: H; x trangdoHo,H] 1acacmatr;!in1Qctu'dngungvai 2kenhlQcth5pva caD. 68 trongdo 2 -2 0 0 0 0 0 0 0 0 2 -2 0 0 0 0 0 0 0 0 2 -2 0 0 A = 0 0 0 0 0 0 2 -2 0 2.fi .fi.fi -.fi -.fi 0 0 0 0 0 0 0 0 .fi.fi - .fi - .fi 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 3.Khaitriln tinhi~udungb(J l(Jc Ma tr?n phan rich Ta co duQcb~ngcach xen ke cac dong cua cac ma tr?n HpH]Ho,...,H]H;-I,H; 1 I... 1 -1 0 0 H=- 1 2J21... 0 0 1 - 1 ... H =~I'" 1 1 0 0 ... 0 J21... 0 0 1 1 ... 11... 1 1 -1 -1 0 0 0 0 HI H 0=- 21... 0 0 0 0 1 1 -1 -1 ... 2 1 I... 1 1 1 1 -1 -1 -1 -1 0 0 ... H]HO = f,..,2'\/21... 0 0 0 0 0 0 0 0 1 1 ... 1 I... 1 1 1 1 1 1 1 1 0 0 ... H3=- 0 2J21... 0 0 0 0 0 0 0 0 1 1 ... 1 Chu6iWaveletrO'ir~c TruonghQpHaar vila xet la mQttruong hQpd?c bit%tcua chu6i Wavelet roi r<:ic.Trang phgnay,tase t6ngquathoakhai nit%mchu6iWaveletroi r<:icvdi lQctn!cchu§'nba't ky. X6tbQlQc2 kenh tn!c chu§'nvdi cac lQc ho,hI, go,gI trongdo hi[n]= gi[-n]. Lam tl!dngtVnhutrlionghQpHaard phgnlIen,tinhit%ungu6ncothe'duQcvie'tla : x[n]=Lx(1)[2k+I]g?)[n- 2] k] +Lx(1)[2k]g61)[n- 21k] ~Z ~Z (3.3.3) Trangdo: x(1)[2k]=(h6])[2]k -/],x[l]) x(1)[2k+1]=(hl(1)[2]k -I], x[/]) \ake'tquaclla phep lQc qua ho,hIcua tin hit%ungu6n x t<:iicac chi s(f chan, vdi 69 3.Khaitriin tinhi~udungbQ lQe he])=h h(1)=h0 0' 'II Tinhi~uxua'td kenhtha'px(l)[2k]l~idu'Qctachthanh2 kenhtha'pvacao,dodothanh phhthlihaid vephaiCl1a(3.3.3)trdthanh: Lx(1)[2k]h~I)[2jk - n]=L xe2)[2k+1]gle2)[n- 21k] +L xe2)[2k]g~2)[n- 21k] ~Z ~Z ~Z (3.3.4) vdi: xe2)[2k]=\hC;2)[22 k -7], x[l]) xe2)[2k + 1]= \hle2)[22k -7], x[l]) Nghlala taapdvng(3.3.3)themmQtl~nmla.g62)[n]la phienbancuaphepIQcd kenhtha'phail~nkethelpvoi la'ym~ulen,nghlala G62)(z) cod~ng: G62) (z) =Go(z)Go(Z2) congle2)[n]tu'dnglingvoi la'ym~ulen,IQcquabangcaor6iquabangtha'p: Gje2)(z)=Go(z)GI(Z2) xetrheathaigian,(3.3.3)cothSdu'QCvietl~ithanhtinhi~ut6nghelptu3kenh: x[n]=2:>(1)[2k+1]g?)[n-2Ik] kEZ +I>e2)[2k+l]gle2)[n-22k]+Lxe2)[2k]g62)[22k-n] kEZ kEZ (3.3.5) Tie'ptvcquatdnh(3.3.5)J l~n, tadu'QCchu6iWaveletrai r~cvoi J batdQ,clingvoi batdQcu6iclingvoiphienbant~ns6tha'p,(3.3.5)trdthanh: x[n]~ ~(~x(j'[2k +l]g?'[ n - 21k] ) +~X"'[2k ]g~J'[n- 2J k](33.6) trongdo, XU) [2k+1]=\h?) [2f k -7], x[l]) xeJ)[2k] =\h~J)[2J k -7], x[l]) j=1,...,J (3.3.7) Trangcongthlic(3.3.6)kS tren,day g?)[n] la phienbanrheathaigiancua(3.3.2), cong6J)[n]la phienbantheothaigiancua(3.3.1). VIffiQiday {x[n]}deli co thSdu'Qckhai triSnthanhchu6irheacongthlic (3.3.7)lien t~pIH/p{g?)[21k - n],gi2)[22k - n],...,gf)[2Jk- n],gf)[2J k -n]} k,nE Z tflo thank m(JtCdsa tTlleehudn eua l2(Z) Netd?ctIlingcuachu6iWaveletrai r~clacachla'ym~u.KS tukenhthli2trddi,m6i b~nhduQcla'ym~uxu6ngbdi2 sovoi kenhtruocdo,vacodQrQnggiaithonggiam xu6ngmQtmh sovoikenhtruoc.Cachla'ym~unayco thSdu'QcbiSudi€n bdihlnh (3.7)vaduQcgQila lily miiulu:dinh,it6: 70 .....- 3.Khaitriln tinhi~udungbQh?c 9 + 10 11 12 13 14 15 16 I I I I I I I . . . mnh 3- 7 Lu'oin~ to'dungtrongchu6iwaveletri'1irl;lc.Trong lllnh la cac 4nh tie'nCliacacham cd sa gif) j =l,J va g?). Cac chii'm tu'dng«ng voi phep lii'u miiu cua chu6i Wavelet. 2.1CaeHnhcha'tcua'chu6iWavelet rO'ir~c : -Tuye'nfink : Vlchu6iWaveletroir~cchibaagdmph§ntichtrongvatichch~pla caeloantU'tuyen tinh,nentuyentinh. (x+y)Cf)[2k +1]=xU) [2k +1]+yU) [2k +1] (x+y)V)[2k]=x(J)[2k]+ yeJ) [2k] -finklien MQth~tho'ngdato'cthu'ongkhongbfftbientheopheptinhlienngang,trongmQth<$ th6ngvoibQlQchaikt~nhIffym~uxuo'ngbdi 2 taco sVbfftbienvoi pheptinhlien ngangffiQtso'channhip.VI v~ycothehinhdungmQtchu6iWaveletroir~cJ-batdQ seba'tbientheopheptinh lien k.2Jnhip,va bien d6i khi tinhlien mQtso'nhipkhong cod~ngk.2J.Hinh ve (3.7)chothffykhi ta tinh lien quamQtbQi so'cua 2J, cac diem tronghioi se changkhit leu chinhno, di~udo khongxay ra khi tinh lien quamQtso' khacbQicuak.2J. M<$nhd~sandaysechungminhl~plu~ncotmhtrlfcquanlIen. M~nhd~3.8 TrangffiQtkhaitri~nchu6iWaveletJbat dQ,neu: x[l]Bxw[2k+l] , j=I,2,...,J x[l- m2J]B xU)[2(k - m2J)+1]thi Changmink: Ne'uy[l] =x[1-m2J)tmcaeM so'khaitri~nchu6iWavelet,heocongthuc(3.3.14)Hi: yU)[2k+1]=(h1W[2jk -I], x[l- m2J]) =(h1U)[2jk-l' -m2J],x[I']) =xU)[2j(k-m2J-i)+1) Lamtu'dngtl,idkenhthftptaseeo: y(J) [2k]=x(J) [2(k-m)] . -Trl,lcgiao: Caehamgf)[n] va gjeJ)[n],j =1,...,Jva cacbansaotinhlien thichhQpt~othanh ffiQt~ptrt!'cchuffn,di~udocodu'Qclavi cachamlIendu'QCxaydvngtirbQIQctrvc chugn,trongdo : 71 0 1 2 3 4 5 6 7 8 I I I I I I I I I g\I) . . . . . g\2) . . . g\3) . . g\4) . g&4) . . . . . . . 3.Khai triln tin hi?u dung bQ 19C (g i [ n - 2k], g] [ n - 21]) = 5[ i - j]5[ k -I] M~nh d~ 3.9 Trangm9tkhaitrienWaveletroir?c,tacocacd~ngthlicvetrt1cgiaosan: (gf)[n - 2J k],g~J)[n- 2J 1])=5[k -I] (g;J)[n - 2]k],g?)[n - 2'I]) =5[i - j]5[k -I] (gf)[n - 2J k],g;f)[n - 2fI]) =0 Changminh: Ta ch(ingminhbhg quyn'ilP Xet 19Cg~])voi biSn d6i Ztu'dngung la G~]).G9i pC])la day tl! tu'dngquancua day g~]), voi biSn d6i z la pu) till : pC;)(z)=G~1)(z)G~;)(_Z-I) VI {go[n]}la tn!cgiaovoi pheptinhtiSnquaffiqts6 chan,taSHYra p(I)(Z)+p(l)(-z) =2 V~ycac M s6b~cchancua pCI)(z) biing0,nga'ilitnl p(1)[O]=1. Do d6, thanhphin daphadiu W~ncua p(1)(z) bhg 1. p(1)(z)c6 thSdu'<;1cviSt la : p(1) =l+z~(1)(z2) GilLsa g~f)[n]tn!cgiaavoi g~f)[n-mi], khid6daytvtu'dngquanc6thSdli<;1cviStla 2f-I . pU) (z) =1+L z' p,U)(Z21) ;=1 vI: G~j+l)(z) = G~1)(z)Go (Z2] ) pU+1)(z) =pU) (z)p(1) (Z2f)nen: =(1+~Zip'(j)(Z2f ))(1+Z2f~(I)(Z2f+I)) Ta dn ch(ingminhthanhphitndaphath(i0 cuap!J+I)(Z)biing1,nghlala cach~s6 f+1 lingvoi Z2 'bhg O.Trangrichd vSphai,tachidn xetrichs6chuathanhphitnda pha. 21-] .. .+1L z' p,U)(z 2])022]~(l)(Z2J ) ;=1 sO'mil cuaZtrongrichs6 lIen c6 d'ilng: l =i +ki +i +mi+1 trongd6i =0, i-I, k,mEZ v~yl khongthSla bqis6cuai+l, dod6: 2]+1 pi+!(z) =1+L Zi p,U+l) (z2f+1) ;=1 tom l<:tibiing quy IWP, ta chung minh du'<;1c: 2] -) . pi(z)=I+Lzip,U)(z2]) )=1,...,1 ;=1 Nghlala g6]) trvc giaovoi g6])[n-ki] 72 3.Khaitriin tinhi~udunghi?[(lC . D~ngthucParseval: Vi t?Ph9P {gf)[n],g?)[n]},j=1,...,Jla tntcchugn,hdnnuabQlQcla khoiphvcloan haonent?Ph9PlIen la d~ydu,t~othanhmQtcdsdtn,(cchugncua12(Z).Dodo,taco d~ngthlicParseval,: 11x[n~I' =&=, (lx(J) [2kf +tIxU) [2k + 1]1'J II.3BQIQCbatdQvacdche'phdngiaidacd'p BQlQcHaarvaSinccod~cdiemtinhi~udu'9Ctachthanh2phienban: phienbantho vaphienbansaibit?tdekhoiphvcchitie'ttuphienbantho.Phienbantholingvai ph~nt~nsr5tha'pcodQphangiaitha'pvaphienbansaibi~tlingvai ph~nta'nsr5cao euatinhi~ungu6n.Ne'utatie'ptvcphantichph~ntinhi~uthot~nsr5tha'pthanh2 kenhvaiclingbQlQCvaapdvngd~quynhi€u bu'ac,tadu'9CmQtcayphanca'pcacdQ phangiai, con gQi la ph~nra da phangiai hay phanra dQphangiai phanca'p (multiresolutiondecomposition).Cdche'phanradaphangiaiconhi€u lingdvngtrong xalytinhit?unhu'lienanh,lientinhi~uVideo. d~t Va=12(Z). ffiQtphantichaaphangidibaog6mmQtdaycackhonggiancondong16ngnhau. Vj c CV2CVICVO Wj+1la ph~nbli tntc giao cua Vj+Itrong Vj, Vj =Wj+1EB Vj+1 J U Vj =Vo j~O giiistY{go[n]}la mQtph~ntU"cua Vasaocho {go[n-l]tz la mQtcd sd cua Va. {go[n- 2k]tz lamQtcd sdcuaVI. b~ngeachd~t : gl[n]=(-It go[-n+1] (*) thlhi€nnhien: (go[n-2k],gl[n-21])=0 nghlal : {gl[~- 21]}IEZC ~ valamQtcdsdcuaWI. V~y: (go[n- 2k],gl[n- 21])k,IEZt~othanhmQtcdsdcuaVa. Tavie't Va=VI EB WI PhantichVIthanhV2EBW2va tie'ptvcquatrinhnaytadu'9C Va=WI EB W2EB ""'" EB Wj EB Vj XetbQlQcbatde)d hlnh(3.5a),lQcphantichla phienbandaongu'9cthai giancualQc t6nghQp. Vlv~y,pheplQcd kenh1,2,...,J+11ala'ytichch?p,chinhla tichvo huangcuatinhi~u gQI taeo va 73 3.Khaitriln tinhi~udunghi) IflC ngu8nvdi dic hamcd s6cua W},Wz,...,WJvaVJ. Tin hi<$usankhi du'Qct6nghQpvdi cac lQc t6nghQp,ta du'Qcke'tqua la hlnh chie'u trQ'cgiaocuatinhi<$ungu8nxu6ngcackhonggianconW1,WZ,...,WJvaVJ. V~ytinhi<$ungu8ndu'Qcphanra thanhphienban tho6 trongVJ va cac chi tie'tthem vao6 Wi, i =1,...J. Tangcllacacthanhphgnthovacacthanhphgnchitie'tthemVaGkhoiphvctinhi<$u ngu8nbandgu. TagQiVJ lakhonggianxa'pxi vaWjla cackhonggianchitie't. Quatrlnhkhoiphvctinhi<$ungu8ndu'Qcthlfchi<$nnhu'san:Tab~tdguvdiphienban thodQphangiaitha'ptrongVJ,sand6b6sungthemchitie'tva tie'ptvcquatrlnhli[lp chode'nkhid?tdu'QcdQphangiaicaDnha'tcu6irung. Vidl,l: Bi) IflC hat di) dung IflC sinc. LQcsincsli'dvngbQlQcg8mlQcnli'arhobangtha'pqua19tu'6ngc6tgnso'dt nIl va lQchobangcaDqualy tu'6ng.KhonggianVo=h(z)co~h6trendo~n(-n,n).Vodu'~c tachdoithanhVI EJjWI vdi VI tren(-nl2,nl2),WI trenkhoangconl~i.VI l?i duQC phantichthanhV2tren(-n/4,n/4)... _V2-' Vl- Vo~ ., ... VIWI'... W2 WI ~... n n n ? 21-1 4" n 2 n (j) ffinh 3-8 Chu6i Wavelet rdi r~edung1gesineco phiin cat ph6 1)'tuf1ng,khong gian Vi duWi+lling vdi ph6 ~ tudngling vdi ph6 [O,nl2!]du<Je phiincat thanh[O,nI2!+1]va[nl2J+\nI2!] tudngling 1ftn11«JtvOiV!+lva WJ+l. K A', A AI THIET KE CAC BO LOC TRfjC CHUAN : Trangphgnnaytasexetde'n2 cachthie'tke'lQctrlfcchu§'n,cachthlinha'tdlfatren phuongphapphantichthanhthuaso',va cachthlihaidlfatrenca'utrucgi~m.Ta se thie'tke'caclQct6nghQp,VIbQlQcla trlfcchu§'n encaclQcphantichc6thesuytu lQctanghQpb;}ngcachdaDnguQcthaigian: hi[n]=gi [-n]. M~nhde3.10 Choh lahamGaplingxungcuamQtlQCvapIa daytVtu'dngquailcuah. p[m]=(h[n],h[n+m]) Giiisli'p tu'dnglingvdi lQcFIR. Khi d6bie'nd6izcuap c6d?ng: 74 3.Khaitriin tinhi~udunghi) l(Jc v~y y'[ZI,Z2]= Ly[n;,n;](ZIZ2rn;(Zlz;}rn~ =Y(ZIZ2,ZIZ;}) Y(ZjZ2'Z}Z;I)=!(X(ZpZ2) +X(-Zp-Z2)) 2 d~t tadu'Qc -I Z2 =ZIZ2Z} =ZIZ2 ; Y(z;,z;) =~(X«z; )1/2(z;Y/2,(z;Y/2(z; rl/2) +X( -(z; )1/2(z; )1/2,-(z; )1/2(z;rl/2): Y(Z Z ) =! (X(ZI/2ZI/2 ZI/2Z-I/2) +X(_ZI/2ZI/2 _ZI/2Z-I/2)) 1'2212'12 12'12 ke'tie'ptaxettru'onghQpH(ym~u1en: [z Z ]= { x[n;,n;] ne'unl =n;+n;, n2=n;-n~ y l' 2 0 ndi khac 1 [ n}+n2 nl - n2 ] A' ' h - h ~ , I ? = x 2 ' 2 neu n},n2clingc an °<;lccunge 0 ndi khac bie'nd6izduQctinhdedang1a: Y(z) =X(ZIZ2'ZIZ;I) ke'tquasankhi 1a'ym~uxudng,sando 1a'ym~u1enSHYtir (1)va (2)1a: { X[nl,n2] ne'un I +n2chan y[npn2]= 0 d' kh'n 1 ac 1 Y(zpzJ =-(X(zpzJ +X(-Zp-Z2»2 [tacoth€ tinhtnfctie'p(3.5.1)tir (*)) hay Thie'tke'bc)1Qcnhi~uchi~u: Vi~cthie'tke'cacht%thdngkhongtachduQCkho khanhdn truonghQpmQtchien.Ta sexet2tru'onghQp: tachduQcva khongtachduQC. 83 (1) (2) (3.5.1) 3.Khaitriin tinhi~udungb(j lQc Nu N N Fez) =an((1- ZiZ-1XI- ZiZ))n (1- ZZiZ-I)n (1- Z;iZ) ;=1 ;=! ;=1 TOngd6 Nu la so'nghi~mtrenvongtrondonvi, N la so'nghi~mbentrongvongtrondonvi, Zli la cacnghi~mtrenvongtrondonvi IZliI=1, Z2i lacacnghi~mbentrongvongtrondonvi IZZil<1va 1/z;; la cacnghi~mn~mngoaivongtrondonvi. 'hztngmink : ~ BiSn d6i Fouriercuap la : P(ej<V)=fp[n]e-j"" =ffh'[k]h[k+n]e-j"" n=-" n=-.,k=., = f fh'[k]h[k +n]e-j""k=-.,n=-., ., ., = L Lh'[k]ej<Vkh[k+n]e-j<V(k+n)k=-.,n=-., ., ro = Lh'Ck]ejd>kLh[k+n]e-j<V(k+n)k=-" n=-" =H'(ej<v)H(ej")=IH(ej"r v~y p[m]=(h[n],h[n+m])~ P(ej<V)=jH(ej<vf BiSnd6izcuap la : p(z)=~>[n]z-n=f fh'[k]h[k +n]z-nn=-.,k=-., ., ., =LLh"[k]zkh[k+n]z-(n+k)k=-.,n=-., =fh'[k](l/ zt Ih[k +n]z-(n+k) k=-ro n=-'" =H.(l/z}H(z) V~y, p[m]=(h[n],h[n+m])~ P(z)=H(z)H.(l/z) . voi It (z)ladathuctheozcocaeM s6la lienh<;fpcuacaeh~s6trongdathuc H(z). Tirrangthuctren,ne'uZkla mQtnghi~mcua P thl 1/z; dIng la nghi~mcua P. Khi h[n]Iiidaythl!c,ne'uZklanghi~mcuaH(z)thl z;,1/Zk,1/z; clinglanghi~mcuaP(z). DoIylu~ntren,P(z) c6 d~.lllg: Fez)=aIT(1- ZliZ-1XI- zliz)fI (1- ZZiZ-I)fI (1- Z;iZ) i=1 i=1 i=1 TllrangthuclIen, ta c6 the tlm H(z), cach lam nay gQi la phan rich thanhthila so'. H(z)duQcgQila thila56'ph6 . Cac thuaso'ph6thl khongduynha't.MQt trongnhung IdjgiiiichoH(z)du'QchQnc6d~ng: N" N H(z) =Fan (1- ZIiZ-I)TI (1- ZZiZ-I) i=1i=! Trangdo,tachQnZ2ila cacnghi~mn~mbentrongvongtron,Zlila cacnghi~mtren 75 3.Khai triin tin hifU dung be?IflC vangtrail,Nula s6nghi~mtrenvangtraildonvi vaN la s6nghi~mbelltrongvang trolldonvi V.l Thie'tke'b{)IQcb5ngcachphfintfchthanhthitas{f: Dungphu'ongphap'phantichthanhthU'as6,tacothethi€t k€ cacbQlQcthoamanmQt s({yetidu chotruck.Debauchiesthi€t k€ hQcaclQcthoadiSuki~nph~ngt6i da (maximallyHatfilters).LQcDebauchiesdaihoi lQckenhtha'pconghi~mbQit6ida t?i OJ=;rr,noicachkhac,daytv Wongquailcobi€n d6iFourierconhiSunghi~mt~i w=;rr(hayt~i~=-1). NgoairaVI P(z) +P(-z) =2 nenP(z)cod~ng: Fez)=(l+z-1Y(I+zYR(z) trongdoR(z) d6i xling (R(z)=R(z-1))va du'ongtrenvangtrail donvi. X6t tru'ongh9P R(z)cob?ct6i thi€u, cacs6h~ngcos6miltU'-k+1d€n k -1 . Khi dovoi rangbuQc tren,tacothetImdu'9CGo(z) Vidu :TimlQcD2tronghQlQcDebauchies Vdik=2lingvoi19ccochiSudai4,P(z) cod~ng Fez)=Go(z)Go(Z~l)=(1+Z-l Y (1+zY R(z) VdirangbuQcb?ct6ithi€u choR(z),R(z) cod~ng: R(z)=az+b+az~l v~y:P(z)=az3+(4a+b)z2+(7a+4b)z+(8a+6b)+(4b+7a)z~1+(b+4a)z-2+az-3 do(*),cach~s6b?cchancuaP(z)b5ng0vap[O]=I,taco : 4a+b=O 8a+6b=1 1 1 a=-- b=- 16' 4 1 1 1 -] R(z) =--z +---z 16 4 16 phantichR(z)thanhthU'as6,tacothechQn: R(z){~-J[1 +v'3 +(1- v'3 )d~+v'3 +(1- v'3 )z] chQncacnghi~md trongvangtraildonvi,setu'onglingvoi lQcbangtha'p: Go(z)=4~(I+Z-1Y[I+./3+(I-./3)Z-I] Go(z)= 1",[1+./3+(3+./3)Z-1+(3_./3)Z-2+(I-./3)z-3]4'\12 (*) hay [ 3]= [ 1+./33+./33-./3 1-./3 :go0.. 4J2' 4J2' 4J2' 4J2 76 3.Khaitrilfntinhi~udungbq h;c g[-2..1]= [ I-J3, _3-J3, 3+J3, _1+J3 ]1 4.[2 4.[2 4.[2 4.[2 Vi du : xet tru'onghQpphlic t<:iPhdn, k =3ling vdi lQcD3co chieudai 6,P(z)co d<:ing: Fez)=(l+z-1Y(I+zY R(z) vdirangbuQcbellct6i thieuchoR(z),R(z)co d<:ing: R(z)=az2 +bz+c+bz-1+az-2 tatinhdu'QcP(z)=(z3+6Z2+15z+20+15z-1+6z-2+z-3Xaz2+bz+c+bz-1+az-2) =az5+6az4+15az3+20az2+15az+6a+az-I +bz4+6bz3+15bz2+20bz+15b+6bz-I+bz-2 +cz3+6cz2+15cz+20c+15cz-1+6cz-2+cz-3 +bz2+6bz+15b+20bz-1+15bz-2+6bz-3+bz-4 +az+6a+15az-1+20az-2+15az-3+6az-4+az-5 =az5+(6a+b)Z4+(15a+6b+C)Z3+(20a+16b+6C)Z2 +(16a+26b+15c)z+(12a+30b+20c)+(16a+26b+15c)Z-1 +(20a+16b+6C)Z-2+(15a+6b+C)Z-3+(6a+b)Z-4+az-5 vip[2k]=0trup[O]=I,taco : { 6a+b=0 lOa+8b+3c=0 giiiih<$,tadu'Qc 6a +15b+10c=1 R(z) =~(3Z2 -18z +38-18z-1+3z-2)128 philntichthanhthuaso',gQia,fJla cacnghi<$mbelltrongvongtrollddnvi, taco: R(z)=A2(1- az-l)(1- fJz-l)(1-az)(I- fJz) R(z) =A2(afJz-2 - (a + 13)(1+afJ)z-1 + 1+(a +13)2 +a2132- (a +13)(1+ap)z +afJz2) d6ngnha'th<$so',ta du'QC: [ A2.afJ=3/128 A 2.(1+afJ)(a +13)=9/64 A2.(1+a 2132+(a+13)2)=19/64 giiiiMphu'dngtrinhtrentadu'Qc: A =1+.Jlo +~5+2.Jlo , 16.[2 a +13=- 8+3.Jlo+6~5+2.Jlo 1+.Jlo +~5 + 2.Jlo afJ =1+.Jlo- 5~5+2.Jlo 1+.Jlo +~5+2.Jlo f a=3/128 b =-9/64 c=19/64 77 3.Khai triin tin hifju dung b(J lQc Vlv~y: Go(z)=(1- Z-1)3A(I- (a +f3)Z-l+af3z-2) =A(I- 3z-1+3z-2- Z-3)(1- (a +P)Z-I +apz-2) =A(I-(3+a+ f3)Z-1+(3(a +f3+1)+ap)z-2 +[3af3- 3(a+f3)-1]z-3 +(3af3+a +p)z-4 - af3z-5) ThaycacgiatrivaotaduQc: [0]=1+J1o +.Js +2J1o go 1612 [1]=5+J1o +3fs +2J1o go 1612 [2]=10- 2J1o+2J5 +2J1o go 1612 [3]=10- 2J1o - 2J5 +2J1o go 1612 [4]=5+J1o- 3)5+2J1o go 1612 go[5]=1+.J1o- J5+2.J1o 1612 .2 Thie'tke' IQc b~ng phlidng phap phan tich thanh thita s6 ki~u gian (lattice factorizations). Matr~ndaphaCl1amQtlQcFIR co chieudai L =2K cod~ng ( Goo(Z) GlO(Z) ] Gp(z)= GOI (z) Gl1 (z) trangd6GI(z)=_Z2k+l1GO(_Z-I) lamatr~nkhongma'thongtin.Ma tr~ntrencothe' dl1qcphantlchthanhthuas6cacmatr~nquaynhu'sau(xem[1]): ( Goo(Z) GlO(Z) ] [n k-I ( 1 ) ] Gp(z)= =Uo I U; GOI (z) Gl1 (z) ;=1 z- Ui = ( c~sai -sinai ]smai cosa; Apd\lllgcacca'utrucphiintlchtren,taco the'thie'tke'bQlQcDz chieudai 4, k =2 matr~nGp la : G (Z)= ( c~sao -Sinao J( 1 - J( c~sal -Sinal ]p smao casaD Z I smal casal ( .. -I .. -I J casaDcasal- smaosmalz - casaDsmal - smaocasalz = sinaocasal +casaosinalz-I -sinaosinal +casaocasalz-I trongd6: 78 3.Khaitriln tinhiiju dungb(}lQc lQckenhtha'pla : Go(z) =Goo(Z2) + z-IGOI (Z2) . =cosaocosal+sinaocosalz-]-sinaosinalz-2+cosaosinalz-3 lQcD2co nghic$mdenca'p2 t;;tiz=-1, dodo : { Go(-I~=casaDcasal- casalsinao- sinatsinao- sina]casaD=0 (1) dGo(e) 1 (=casalsmao+2smatsmao+3smalcasaD=0 2) dO) 0) =:rc (1)tuongduongvoi : cos(ao+a])- sin(ao+al) =0 hay ao +a] =k:rc+:rc4 VI: Go(1)=J2 nen: :rc ao+al ="4 thayvao(2)tadU9C: :rc :rc a =- a =-- 0 4'] 12 Go(z)tra thanh : Go(z)=cos(:rc/3)cos(-:rc/12)-sin(:rc/3)sin(-:rc/I2 )Z-I - sin(:rc/3)sin(-:rc/I2 )Z-2 +cos(:rc/3)sin(-:rc/I2 )Z-3 Go(z) = 1",(1+.J3 +(3- .J3)Z-1+(3- .J3)Z-2+(1- .J3)Z-3)4,,2 D6chinhla lQcDebanchiesD2ta da timdU9Cd tIeDb~ngphuongphapphanrich thanhthuasd. BOLOC NHIEU CHIEU: U'ymill trongkhonggianmchit~u: La'ym~utrongkh6'ngianmchiendU9Cth1!chic$nhokhainic$mgian. Djnhnghja3.3 Gian: ChoaI,a2,...,amla n vectodQcl~ptuyentinh.T~ph9PS ta'tcacact6 h9PtnyentinhcuaaI,a2,...,amvoicachc$sdngnyendU9cgQilamQtgian. GQiD la matr~nvoicacvectocQtlaaI,a2,...,amthlgianS noitIeDCOd;;tng: S ={Dk / k E Zm} tanoiD la matr~nxacdinhquatrinht;;tOmill, haymatr~nbiendieDgian. Tangquat,tanoit~ph9Pcacvectok la giannh~p(trongtru'ongh9PgiantIeD laZm)vat~ph9PcacDk lagianxna't. Giantachd~(C!c: lagiancothedU9CbiendieDb~ngmQtmatr~nduongcheo. M(}te'baadanv,i: la t~ph9PcacdiemsacchohQicacgianxna'tinhlienden 79 3.Khaitriln tinhi~udungbi?lQc ta'tcacacdiemtrongte'baat~othanhgiannh~p.so'phgntd'trongmQte'baa donvi biendi~ndQthu'acuaquatrinht~omfiuvadu'Qcxacdinhbiing N =del (D) MQtcosetla ke'tquacuapheptj1lhtie'nmQtgianxua'tde'nmQtdiemba'tky trongte'baa donvi. Ne'ute'baa donvi co N =del (D) phgntd'thl co dungN cosetphanbi<$t,hQicuaN cosetnaykhoiph\lcl~igiannh~p. Mi?tgiandaocuamQtgianco matr~nbiendi~nD la gianco matr~nbien di~n Dr=(D-l)T Mi?tte'baaVoronoila te'baadonvi cocacphgntd'ggnnha'tvdi tam.Ne'utin hi<$udn t~omfiutrongmientgn so'cobangthongbi ch~ntrongmQtte'baa donvi, cacph6sekhonghungHiplennhauva tinhi<$uco thedu'QCph\lch6i titcaemfiudu'Qct~o. D2 ~ D. DI Im . I -A +......-« , y (a) (b) ffinh 3-9 Hai cii'utruegiim thuongg~p.(a) Uiy m~uMi 2 trongkhonggian2ehi~u.(b)Uiy m~uhlnhngfidi~m .1.1La'ym§utrongkhonggianmchi~u: La'ymfiuxu6ngconghlala cacdiemtronggiandu'Qcgill l~i,boquacacdiemconl~i. GQiD la matr~nbiendi~ngian,la'ymfiuxu6ngmQtdayx : zm-+R sedu'Qcdayy nhu'san: y [II]=x[DII] TrangmienFourier yew)=~ :LX((Dtr1(w - 2n-k))N kEUt c nE Zm N =del(D) wla mQtvectorthtfcmchien II ,k la cacvectornguyenmchien Iffym~ulenmQtdayxsedu'Qcdayynhu'san: y[n]= { X[D-ln] ne'~n~Dk0 notkhac TrangmienFourier: vdi 80 :.Khaitriln tinhifU dungbq lflC Y( m)=XeD I m) .re'uphepla'ym~uco gi~llltu'ongungtachdu'QC,tanoi phepla'ym~ula tachdu'Qc. If d1,1: ~rongphgnnay, taxet2 tru'onghQpla'ym~utachdu'QCva khongtachdu'QC. Ii dl:l1 ..Lity mdu tach du:(/c ,a'ym~uxu6ng(len)bdi2 trongkhonggian2 chieu.quatrlnhla'ym~ucothebieu li~nb~ngmatr?n: Ds=(~ ~)=21 gbaadonvi gdmcacdiem: (nl, n2)E {(O,O), (I,O), (0,1), (I,I)} rangmienz,cacdiemtrentu'ongungvdibie'nd6iz la : {I -1 -] -] -j },Zj,Z2,ZjZ2 gbaoVoronoico4phgntii'tu'ongungvdibl?lQccoN=det(D)=4kenh ,a'ym~uxu6ngsedu'qcdayke'tqua: y[nl , n2] =x[2nl ,2n2] rangmienz,bie'nd6izsela : Y[ZI,Z2]=:(X(ZI~'Z~)+X(zl~,-zh+X(-z1,z~)+X(-Zl;,z~)J ~gu'qcl?i, la'ym~ulen : { [ nj n2 ] A" .:;- [ J - x -,- neunj,n2chany npn2 - 2 2 0 noikhac :cobie'n d6izla: Y(Zl , Z2)=X ( Zj2 ,z;) ie'nd6iZsaukhila'ym~uxu6ngbdi2,rdi la'ym~utenbdi2 sela : 1 Y(Zl,Z2)=-(X(ZI,Z2)+X(-ZI,Z2)+X(ZI,-Z2)+X(-ZI,Z2» 4 (dngungvdi { X[nj, nJ ne'un]>n2chg-n y[npn2J= 0 .kh'nOI ac id!l2..Liiymdukh6ngtachdu:(/c.. ettru'onghqpla'ym~utheohlnhngudiem,daylaca'utruct?Om~uhaichieukhong ~chduqc.Giantrongtru'onghqpnaydu'qcbieudienb~ngmatr?n: DQ :::J( 1 1 )1 -1 '\detDQ=2nenbl?lQctu'ongungselabl?lQc2kenh. laconMnxetlamatr?nDQkhongduynha't,vi dl,lmatr?n: 81 3.Khai trii'n tinhi~udungb(ih}c , ( 1 1 )DQ= -1 1 clingbieudi~ndu'c;Jcquatdnhla'ym~uhlnhngtidiem] Te'b~lOVoronoichogi~illnglidiemla me)thlnhthai(hlnhvuongnghieng)vagiandao clingV?y.Do do,co theapdvngdemahoahlnhanhbdiVIne'ugioih~nvaGmi€n naythl(a)ph6cuatinhi~uva cacphienbanI~pI~ido t~om~usekhongd~mten nhauva (b)miitngu'oikhongnh~yvoi de)phangiaidQcrheadu'ongcheonencothe chQnbe)IQcbangtha'pdt botheoph§ndu'ongcheo. Haidiemtrongte'baadonvi la : 110=(~) 111= (~) tu'c5nglingvoibie'nd6iz trongmi€n z la I va Z;I haicosetu'onglingla2 t?Phc;Jp: {(nl,nz)/ nl +nzchan} va {(nl, nz)/nl +nzIe } La'ym~udu'oirheaca'utrucIa'ym~ungudiemsedilqcke'tqua: y[nl,nz]=x[nl + nz,nl - nz] Betlmcongthliclienh~trongmi€n z , d~ty' ladaynhilsail: , { X[nl,n2] ne'un1+n2chan y [nl,n2]= 0 ,.( I ?neu nl +n2e thlbie'nd6izcuay' la : y'(zpzJ= L y'[npn2] z;nl z~n2 (111,112)EZ2 = L x[ill' n2] z~nl z;n2 nl+n2chan - I [ " [ ] -111 -112 " -"2 L... x npn2 Zl Z2 + L... I =-(X(Zl' Z2)+X(-Zl ,-Z2))2 m~tkhactacothevie't: Y'(ZpZ2)= L x[""n,](-1)""'z;' z~,] (*) [ ] -IIJ. -112-" [ ' ' ] -..;-~ -1IJ.+112X npn2 Zl Z2 - L... Y npn2 Zl Z2 nl +n2chan trongd6 n;,n~la2 s6nguyenthoa D(:}(::) nghlala n] =n1+n2 , , n2=n] - n2 82 3.Khai triln tin hi?u dung bQ h,Jc ',2.1Tru'O'nghQ'ptach du'Q'c: -1t 1t f] x HL LH ngang LL LL LH HL HH.~~mdqc (a) (b) Hinh 3-10 Bt) lctachdu<;1chai chi~u,voi phep Iffy mb xu6ngtachdU<;1cbdi 2. (a) Phan di theochi~ungangr6i theochi~udc.(b)Phan chiaph6 trongmi~ntiin so'. Tadungca'utruccascade,vi dVsauset<:\ora4 IQcclingkichthu'dc,phatuye'ntinhva tn!cgiao,trongdo2lQcdo'ixungva2lQcphando'ixung(xemtrong[1]) H"(z,,z,) =[O,R,V(Z"Z,)]So D la matr?nlamtr€ codu'ongcheola 1,Z;I,Z;I,Z;IZ;I Ri vaSola cacmatr?ntvado'ixung(persymmetric),nghlalaRithGa R=JRiJ (3.5.2) phuongtr'inh(3.5.2)cungvdiyelldu Ri lamatr?nUnitasechophepthie'tke'cacIQc cophatuye'ntinhva trvcgiao(trongtru'onghQp1 chi€u kh6ngtheco IQCvila pha tuye'nti hvilatrvcgiao). VdiK thOatvado'ixungvaUnita,Ri cod<:\ng: trongdo I ( 1 J ( 1 1 J ( R2i J ( 1 1 ) ( 1 JRi ="2 J 1 - 1 R2i+1 1 - 1 J R2i,R2i+lla matr?nquayca'p2 x2va s = ( Ro J ( 1 1 J ( 1 )0 RJ 1 - 1 J 2Thie'tke'IQckhongtach du'Q'c,tru'O'nghQ'pngii diim : Trangphftnay,tases\i'dvngca'utruccascadedet<:\oracacbQIQc,cophatuye'ntinh ho~ctnfcgiao.Tu'dngtv nhu'tru'onghQp 1 chi€u, ne'uIQc thGadi€u ki~nkh6i phvc loanhaGthlmatr?ndaphaco thedu'QCthuaso'hoatheoca'utruccascadesau: J ( 1 0 J ( 1 0 J Hp(ZJ,Z2)=OR2i 0 -I RJi 0 ;-1 Rol=k-I Z2 z, dElQcophatuye'ntinh,Rjiphaila matr?ndo'ixung,deIQCla trtfcgiaotaphaicoRji lamatr?ll U nita. (3.5.3) 84 3.Khai triln tin hifU dung b(J 19C XettntonghQptn!cgiao,tasetlmlQcphongtheotru'onghQp1chiSucilaOaubechies. TasetlmlQcconghi~mzerode'nb~ccaDnha't~i(-1,-1) Trudche't,xet(3.5.3)d tru'onghQpdongiannha'tk=2,matr~nHpcod~ng: H/zpzJ =rl [ R2i ( 1 ~I J Rli ( 1 ~I J] Ro i=k-I 0 Zl 0 Zi =R" (~ z~}" (~ z~,)Ro Rijlacacmatr~nUnitanenlamatr~nquay,cod~ng: ( cosaz -Sinaz J( 1 0 J( cosal -Sinal)H (ZI,zZ)= . .p smaz casaz 0 ZJ-I smaJ casaJ (1 ~]Yc~sao -sinaoI 10 ~~ SlOOp cosOp) = l(c~saz -~:-Isillaz ) (c~sa] smaz Zzcosaz lsmal ( cosao smao )smao casaD ( ~.. casazcasa] - Zz smazsma] - sinazcasal +Z;Icosazsina] ( c~sao - sinao )smao casaD -ZI-I sinal ) -I z] cas a] -] . -ZI cosaz smal -I' . -ZI smaz smal -I -I . ) -zz ZI smaz casal +Z;IZ;I cosaz casal H00(ZI' Z2) =casa2casal casaD- Z~Isina2sinal sinao -1 . . -1 -I . . - ZI casa2 sma] smao - Z2Zl sma2 casal smao HO] (ZpZ2) =-casa2casalsinao+Z~Isinazsinalsinao -I' -] -I . - ZI casa2 smal casaD- Z2ZI sma2 casal casaD HIO(ZpZ2) =sina2casalcasaD+Z~Icasa2sina] casaD -I' . . -] -I . - Z] sma2sma] smao +Z2Z] casa2 casal smao Hli (zp Z2)=-sin a2casal sinao - z~]casa2 sina] sinao -].. -] -] - z] sma2sma] casaD+Z2Z] casa2 casal casaD 85 3.Khuitriln tinhi~udungbQltlc lQcchabangthc1pquala : Ho(zj,zz) =Hoo(zjzZ,Zjz~j)-z;jHoj (ZjZZ,Zjz~j) -j . . =casazcasal casaD-Zj Zzsmazsmaj casaD -j -j' -z . . -ZI Zz casazsmajcasaO-zj smazcasajsmao -I . -z. . . -Zj casazcasajsmao+zj zzsmazsmajsmao -z -j' -3 . -Zj Zz casazsmajcasaO-zj smazcasajcasao V~y,hocod?ng: [ ... . . -smaz smajcasaD smazsma]smao hJnj,nz]= casazcasajcasao -casazs~najs~ao -sinazc~sajsinao -casal smajsmao -casalsmajcasaD [ tgaj =sinazcosajcosao-catgaz catgaztgao catgazlgajtgao - sina, COga, cosa" J - tgajtgao tgao catgaztgaj I] D~t Uo=tgao, UI=tgaI, U2= catga2, . , c =-smazcasalcasaD thllQcbangthc1pcod?ng; ho[n,.n,]~ {-a, aj aoaz -uDal ao IJ (3.5.4) aoajaz ajaz lamtu'ongtv lQcquabangcaDla : Hj (Zj, zz) =HJO(ZjZZ,Zjz~])+zj-IHll (ZjZZ'Zjz~j) . -j' =smaz casal casaD+Zj Zzcasal smal casaD -z -1' . . -z . - Zj Zz smazsmaj smao +Zj casal casal smao -1 . . -z .. -Zj smaz casal smao - Zj Zzcasal smaj smao -z -j.. -3 - Zj Zz smaz smaj casaD+Zj casal casal casaD V~y,hIcod?ng; , h, [n, . n,]=[~ 1 ={-l - catgaztgaj tgao tgajtgao catgaztgajtgao - catgaztgao tgaj - cotga, } (3.5.5) - aoaz ao aoajaz - aoaz -a, }ajao aj tacotheviet l?i nhu'sau: H ( ) [ -I -z -] -z -3 ]0 Zj,ZZ =c a1zj Zz -aoajzl Zz -az +aoazzj +aozj +ZI Lamtu'ongtv nhu'lQCOaubechies trong tru'ongh<;$p1 chi€u, ta tim lQc co zero t?i z = 86 /ra;triln tinhifU dungbe?[fie 1,-1)Mn b~ck-I .Trang tru'onghQpk =2,taco Ho(-l,-l)=0 (1) ~o(-1,-1)=0 (2)] :° (-1,-1)=0(3)2 H ( ) ( 1 -] -2 -30 7],Z2 =C - +aDz] -aOa2z] -a2z] -] -2 -] -] -2 -] )- aOa2z] Z2 + aOa]a2z] Z2 +aOa]z] Z2 - a]z] Z2 8H (0 -2 -3 -2 T=c-a]z] Z2+2aoa]z] Z2 -aOa2z]] 2 -3 3 -4 -2 -] 2 -3 -] )- aDz] - z] - aOa]a2z] Z2 - a]a2z] Z2 8H ~ ) 0 -] -2 -] -2 -2 -2 -=ca]z] -aOa]z] -aOa]a2z] Z2 -2a]a2z] Z2 &2 'ongtrlnhtrentu'ongdu'ongvdi : iii { , a]+aoa]-a2 -aoa2+ao-I+aoa]a2-a]a2=0 a~+2:oa] - aoa2+2:0 - 3+aoa]a2- 2a]a2-=0 a] aoa] + aoa]a2 a]a2 - 0 { aJI +ao+aoa2-a2)- a2-aoa2+ao-1 =0 (4) hay,a](1+2ao+aoa2- 2a2)+2ao-aoa2-3: 0 (5)-a](I+ao-aoa2+a2) - 0 (6) I)taco: al=0guyra { a] =2- ao 2ao-aoa2-3 =0 1+ao- aoaz+az=0 guyra ao+1 a2=- ~-,~ !hayvao(5)ta QuQc: 2ao- aoa2- 3 - ag+6ao- 3 a] = = 2 1+2ao+aoa2- 2a2 3ao- 2ao- 3 thayvaophu'ongtrlnh(4) tadu'Qc: 2 ' )( ( ) ao+1 ) 2 )( ( ) ao+1 )(-ao +6ao- 3 1+ao+ ao-1 ao-1 +(3ao- 2ao- 3 ao-1 - ao+1 ao-1 =0 ( -4a J (-ag +6ao-3)(2ao +2)+(3ag-2ao -3) ~ =0 ao-1 (-ag+6ao-3)(ag-1)-2ao(3ag-2ao-3)=0 hay { ao=J3 a2=2-J3 ho~c { ao=-J3 a2 =2+J3 87 '.Khai triin tin hi~udung bQ lflC a4 - 2a2 - 3 = 00 0 (a~+I)(a~- 3)=0 ao =IFJ ao =+FJ a--Tj0 - -V.:J roml<;tita du'Qccac nghit$m: (nl)ao=+J3, a] =0, a2=2-J3,tgao=+J3,tga]=0, catga2=2-J3 (n2) ao =-J3, a] =0, a2 =2+J3, tgao =-J3, tga] =0, catga2 =2+J3 (n3) ao = +J3, a] = -J3, a2 = 2 + J3, tgao = +J3, tga] = ~J3, catga2 = 2 + J3 (n4) ao =-J3, a] =+J3, a2 =2-J3, tgao =~J3, tga] =+J3, catga2 =2~J3 rdi rdi II (n!) ta co ta co : a2=2+FJ a2 =2-FJ a] =-FJ a] = +FJta co : I+FJ. -+- casal =II , sma2- - 2..[2 I casaD =I- , 2 ke'thQpvdi dieu kit$n G(+l,+l) = ..[2 , ta co . I+FJ c = -SIll a2 cas a] casaD = - 412 "I . I-FJ tu(n2)taca casaD =I- , casal =II , sma2 =I~ 2 2-v2 I . I-J] casal =I"2 ' sma2=I 2..[2 I . I-FJ casal=I"2 ' sma2=I 2..[2 I+FJ SUYfa c=-- 812 Thaycacgia tri vaG(3.5.4)va(3.5.5)tadu'QCcacI9Cho,hI- 88 I-FJ suyfa c = 412 I tli(n3)taco casaD=I- , 2 I-FJ suyfa c=- 812 tit(n4)taco 1 casaD=I- , 2